Hausken, K. (2000) Migration and intergroup conflict.
Economics Letters , 69(3), pp. 327–331
Link to official URL: http://dx.doi.org/10.1016/S0165-1765(00)00326-8
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Migration and intergroup conflict
Kjell Hausken*
Abstract
Two groups in conflict produce and appropriate internally generated consumable output in a two-stage game
assuming equal within-group sharing and endogenous group sizes. It is shown how agents leave groups with
high productive efficiency and migrate to groups with high appropriative and defensive capabilities.
Keywords: Within-group strategic choice; Allocation of endowment; Production; Appropriation; Free-riding; Between-group
competition; Group decisiveness; Intergroup migration
If you can’t beat them, join them. If you can’t join them, beat them. What is the underlying
principle by which agents decide to beat or join groups? This article answers the question allowing
groups to differ w.r.t. productive efficiency, appropriative and defensive capability, allowing varying
degrees of decisiveness in between-group competition. Assuming intergroup migration, this article
extends two-level conceptions within three fields.1 Agents often prefer to produce consumable output,
but may have several reasons not to do so. First, production is costly. Second, produced output may be
appropriated. Third, an agent may prefer to appropriate rather than to produce. Fourth, the agents may
prefer to free-ride. These reasons are problematized when agents are allowed to migrate between
groups.
1
The first is collective rent seeking (Katz et al., 1990; Baik and Shogren, 1995; Hausken, 1995a,b, 1998; Baik and Lee,
1997; Lee, 1995; Nitzan, 1991a,b, 1994). The second is the analysis of the impact of product-market competition on
managerial slack (Hart, 1983; Horn et al., 1995; Tirole, 1988; Vickers, 1995; Winter, 1971). The third involves conflict
between actors (Grossman, 1991; Grossman and Kim, 1995; Hausken, 2000; Hirshleifer, 1995; Neary, 1997; Skaperdas,
1992; Skaperdas and Syropoulos, 1997; Usher, 1992; Usher and Engineer, 1987; Noh, 1998).
Consider two groups in competition. Agent j in group i is endowed with an initial resource
endowment r i which may either not be allocated (free-riding, leisure), or may be allocated w ij to
production and s ij to appropriation and defense (appropriation for short), 0 # w ij 1 s ij 5 r i , i 5 1,2.
Two groups with sizes n 1 and n 2 produce consumable output (products, goods, outcomes, prizes,
benefits, rewards, payoffs)
n1
B1
Ow
n2
1j
1 B2
Ow
2v
,
v51
j51
where B1 and B2 specify how efficiently output is produced. Applying the conventional ratio form
(Tullock, 1967) to determine each group’s and each agent’s ability to appropriate output, agent j’s
payoff in group 1 is 2
P1j (s 1j ,S
21j
SO
D
D
n1
m
F
n
G
n
O
O
1
2
F1
1
j51 s 1j
) 5 ] ]]]]]]]]]]
B
(r
2
s
)
1
B
(r 2 2 s 2v ) ,
n1
m
n2
m
1
1
1j
2
n1 F
j 51
v51
s
1F S
s D
SO
1
j51 1j
2
O
v51 2v
(1)
21j
where S
is the set of all strategies by all agents in the two groups except agent j, m is the
between-group decisiveness which specifies between-group sharing, and F1 and F2 specify how
effectively output is appropriated (and defended). Allocation into production leads to enlargement of
the size of the pie of output produced by the two groups, while allocation into appropriation increases
the share of the pie accruing to each group.
We analyze a two-stage game, solve each agent’s maximization problem and check when
Pij (s ij ,S 2ij ) . r i to avoid free-riding. In the first stage agents decide which group to belong to,
dependent on which group gives the highest payoff, suitably taking into account how the agents
allocate their endowments between production and appropriation in the second stage. In so doing,
each agent takes all the other agents’ group membership decisions as given. Acknowledging
equivalent agents, maximizing the aggregate group payoff is equivalent to maximizing the individual
payoff in a symmetric equilibrium where each agent receives the same payoff. In so doing, agent j
takes suitably into account how the agents allocate their endowments between production and
appropriation in the second stage. In the second stage the agents make their choices simultaneously
and independently, taking the group sizes n 1 and n 2 as given. Agent j in group 1 takes the production
versus appropriation allocation of the other agents in group 1 as given, and also takes the production
versus appropriation allocation w 2v versus s 2v of the agents in group 2 as given. He then chooses s 1j to
maximize his payoff. We first consider the second-stage decision. Setting the derivative of
P1j (s 1j ,S 21j ) in (1) w.r.t. s 1j equal to zero gives
SO
D
O
O
1
2
≠P1j (s 1j ,S 21j )
F2S v51 s 2vD
1 F1 m
j 51 s 1j
]]]] 5 ] ]]]]]]]]]]]]
B
(r
2
s
)
1
B
(r 2 2 s 2v )
n1
m
n2
m 2
1
1
1j
2
≠s 1j
n1 F
j51
v51
s
1F S
s D
S SO
n1
m21
n2
m
F
n
O
D
D
F SO s D
1
1 ] ]]]]]]]]]][2B ] 5 0.
n F SO s D 1 F SO s D
1
j51 1j
v 51 2v
2
n1
1
1
2
n1
1
j51 1j
n
O
G
m
j51 1j
m
n2
2
m
1
(2)
v51 2v
It is straightforward to endogenize the within-group sharing rule and show that egalitarian sharing is an equilibrium. See
Noh (1998) for a fuller treatment of within-group sharing rules.
In a symmetric Nash equilibrium identical agents devote the same amounts w 1j 5 w 1 and s 1j 5 s 1 to
production and appropriation, respectively. (2) simplifies to
(n 1 s 1 )m11 F1 B1 1 m(n 2 s 2 )m11 F2 B2 1 (m 1 1)n 1 s 1 (n 2 s 2 )m F2 B1 2 F2 m(n 2 s 2 )m (n 1 r 1 B1 1 n 2 r 2 B2 )
5 0.
(3)
Multiplying (3) by F1 (n 1 s 1 )m /F2 (n 2 s 2 )m and subtracting from that version of (3) where the indices 1
and 2 are permuted, gives
S D
s
n F2 B2
]1 5 ]2 ]]
s 2 n 1 F1 B1
1 / (m11 )
,
(4)
mF 12 / (m11 ) (n 1 r 1 B1 1 n 2 r 2 B2 )
]]]]]]]]]]]]]]]]]
s1 5
1 / (m11 )
1 / (m 11) m / (m11 )
1 / (m11 ) m / (m11 ) ,
n 1 (m 1 1)B 1
(F 2
B1
1F1
B2
)
(5)
mF 11 / (m11 ) (n 2 r 2 B2 1 n 1 r 1 B1 )
s 2 5 ]]]]]]]]]]]]]]]]]
,
n 2 (m 1 1)B 12 / (m11 ) (F 11 / (m 11) B 2m / (m11 ) 1 F 21 / (m11 ) B 1m / (m11 ) )
F 11 / (m11 ) B 2m / (m11 ) (n 1 r 1 B1 1 n 2 r 2 B2 )
P *1j 5 ]]]]]]]]]]]]]]]
,
n 1 (m 1 1)(F 12 / (m11 ) B 1m / (m11 ) 1 F 11 / (m11 ) B 2m / (m11 ) )
(6)
F 21 / (m 11) B 1m / (m11 ) (n 2 r 2 B2 1 n 1 r 1 B1 )
P *2v 5 ]]]]]]]]]]]]]]]
.
n 2 (m 1 1)(F 11 / (m11 ) B 2m / (m 11 ) 1 F 21 / (m 11) B 1m / (m11 ) )
In the first stage, agent j chooses groups 1 or 2 to maximize his payoff. Agent j is indifferent w.r.t.
group membership when P *1j 5 P *2v . Applying (4), (6), and n 1 1 n 2 5 N, gives
S D S D
S D
S D
s
n F2 B2
]1 5 ]2 ]]
s 2 n 1 F1 B1
n
]1
n2
m11
1 / (m11 )
F1 B2
5] ]
F2 B1
n2
5 ]
n1
S D S D
(m21 ) / m
F
]2
F1
1/m
n2
5 ]
n1
2
S D
B
F
]2 5 ]2
B1
F1
S D
2 / (m11 )
B
]1
B2
(m21 ) / (m11 )
,
(7)
m
,
(8)
F 11 / (m11 ) B 2m / (m11 )
n 1 5 ]]]]]]]]]]]
N,
F 11 / (m11 ) B m2 / (m 11) 1 F 21 / (m 11) B 1m / (m11 )
F 21 / (m11 ) B 1m / (m11 )
]]]]]]]]]]]
n 2 5 1 / (m11 ) m / (m 11)
N,
F2
B1
1 F 11 / (m 11) B 2m / (m11 )
n 2 m(n 1 r 1 B1 1 n 2 r 2 B2 )
s 1 5 ]]]]]]],
n 1 (m 1 1)NB1
n 1 m(n 2 r 2 B2 1 n 1 r 1 B1 )
s 2 5 ]]]]]]],
n 2 (m 1 1)NB2
(9)
n 1 r 1 B1 1 n 2 r 2 B2 F 11 / (m11 ) B 2m / (m 11) r 1 B1 1 F 21 / (m11 ) B 1m / (m11 ) r 2 B2
P *1j 5 P *2v 5 ]]]]] 5 ]]]]]]]]]]]]]]
1 / (m11) m / (m11 )
1 / (m11 ) m / (m11 ) .
(m 1 1)N
(m 1 1)(F 1
B2
1F2
B1
)
(10)
Free-riding is avoided when Pij (s ij ,S 2ij ) . r i , which gives
1 / (m11 )
m / (m11 )
F1
B2
(n 1 r 1 B1 1 n 2 r 2 B2 )
]]]]]]]]]]]]]]]
. 1,
1 / (m11 ) m / (m11 )
n 1 r 1 (m 1 1)(F 2
B1
1 F 11 / (m11 ) B 2m / (m11) )
(11)
F 12 / (m11 ) B 1m / (m11 ) (n 2 r 2 B2 1 n 1 r 1 B1 )
]]]]]]]]]]]]]]]
. 1,
n 2 r 2 (m 1 1)(F 11 / (m11 ) B 2m / (m11 ) 1 F 21 / (m11 ) B 1m / (m11) )
for fixed sized groups and
S D
n r
n r
B 1 ]]B . (m 1 1)S1 1 ]D],
n r
n r
n2r2
n2
B1 1 ]]B2 . (m 1 1) 1 1 ] ,
n1r1
n1
2 2
1
1 1
2
2
2
1
1
(12)
for intergroup migration. An agent in group 1 prefers intergroup migration rather than fixed sized
groups when (10) is larger than P *1j in (6), i.e.
S D
F
]1
F2
S D
1 / (m11 )
B
]2
B1
m / (m11 )
n1
, ].
n2
(13)
The result in (4) is well known in the literature (Hirshleifer, 1991; Grossman and Kim, 1995;
Skaperdas and Syropoulos, 1997). The results in (5)–(13) are not known and can be summed up in
nine points. (1) If two groups can agree on equivalently increasing B1 5 B2 , the equilibrium mixture
of allocation into production and appropriation remains unchanged, although their payoffs increase.
(2) Increasing B1 in group 1 causes higher productivity and payoffs in group 1, but causes
considerably more appropriation and even higher payoffs in group 2. (3) Equivalently increasing
F1 5 F2 in the two groups does not alter the equilibrium and the payoffs. (4) Increasing F1 in group 1
causes higher productivity and payoffs in group 1, and more appropriation and lower payoffs in group
2. (5) Increasing decisiveness m causes larger allocation to appropriation and lower payoffs. (6) The
ratio of the payoffs in groups 1 and 2 is inversely proportional to the ratio n 1 /n 2 of the group sizes,
proportional to F1 /F2 (in a manner approaching independence as m increases), and inversely
proportional to B1 /B2 (in a manner that approaches linear dependence as m increases). (7) Allowing
intergroup mobility when B1 . B2 causes migration to group 2 in a manner that becomes more
pronounced when m increases, and moderately large (and equivalent) payoffs in the two groups. (8)
Allowing intergroup mobility when F1 . F2 causes migration to and more production in group 1 in a
manner that becomes less pronounced when m increases, very high appropriation in group 2, and
higher payoffs. (9) Intergroup migration causes the ratio s 1 /s 2 of allocation into appropriation to be
inversely proportional to F1 /F2 , inversely proportional to B1 /B2 when 0 # m , 1, and proportional to
B1 /B2 when m . 1.
The significance of the results lies in the non-trivial implications of the model for resource
allocation (division of labor) for each agent, welfare between groups, intergroup migration, and
adjustment of group size. Central to the model is the placement of consumable output in a common
pool. This creates a benchmark for individual and group behavior where property rights are
determined (Neary, 1997) by each group’s ability to appropriate from the common pool. Determining
property rights by other factors, e.g. closeness to production or judicial criteria for ownership, and
allowing appropriated output not to be 100% exploitable (Grossman and Kim, 1995), suggest an
opposite benchmark where appropriation is absent. The former benchmark causes agents up to a point
to beat rather than join the group with higher productive efficiency. A group may cause movement
toward, without reaching, the latter benchmark by increasing its appropriative capability, encouraging
the other group to increase its productive efficiency, decreasing the between-group decisiveness, or
forbidding emigration (implies higher payoff to the other group).
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